## The Annals of Probability

- Ann. Probab.
- Volume 22, Number 1 (1994), 17-27.

### Large Deviations, Moderate Deviations and LIL for Empirical Processes

#### Abstract

Let $(X_n)_{n\geq 1}$ be a sequence of i.i.d. r.v.'s with values in a measurable space $(E, \mathscr{E})$ of law $\mu$, and consider the empirical process $L_n(f) = (1/n)\sum^n_{k=1} f(X_k)$ with $f$ varying in a class of bounded functions $\mathscr{F}$. Using a recent isoperimetric inequality of Talagrand, we obtain the necessary and sufficient conditions for the large deviation estimations, the moderate deviation estimations and the LIL of $L_n(\cdot)$ in the Banach space of bounded functionals $\mathscr{l}_\infty(\mathscr{F})$. The extension to the unbounded functionals is also discussed.

#### Article information

**Source**

Ann. Probab., Volume 22, Number 1 (1994), 17-27.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988846

**Digital Object Identifier**

doi:10.1214/aop/1176988846

**Mathematical Reviews number (MathSciNet)**

MR1258864

**Zentralblatt MATH identifier**

0793.60032

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F10: Large deviations

Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60G50: Sums of independent random variables; random walks

**Keywords**

Large deviations moderate deviations law of iterated logarithm (LIL) isoperimetric inequality Smirnov-Kolmogorov theorem

#### Citation

Wu, Liming. Large Deviations, Moderate Deviations and LIL for Empirical Processes. Ann. Probab. 22 (1994), no. 1, 17--27. doi:10.1214/aop/1176988846. https://projecteuclid.org/euclid.aop/1176988846